Theorem filconn 24217

Description: A filter gives rise to a connected topology. (Contributed by Jeff Hankins, 6-Dec-2009.) (Revised by Stefan O'Rear, 2-Aug-2015.)

Assertion

Ref	Expression
filconn	⊢ (𝐹 ∈ (Fil‘𝑋) → (𝐹 ∪ {∅}) ∈ Conn)

Proof of Theorem filconn

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		id 23	. . 3 ⊢ (𝐹 ∈ (Fil‘𝑋) → 𝐹 ∈ (Fil‘𝑋))
2		filunibas 24215	. . . 4 ⊢ (𝐹 ∈ (Fil‘𝑋) → ∪ 𝐹 = 𝑋)
3	2	fveq2d 6900	. . 3 ⊢ (𝐹 ∈ (Fil‘𝑋) → (Fil‘∪ 𝐹) = (Fil‘𝑋))
4	1, 3	eleqtrrd 2867	. 2 ⊢ (𝐹 ∈ (Fil‘𝑋) → 𝐹 ∈ (Fil‘∪ 𝐹))
5		nss 3977	. . . . . . . 8 ⊢ (¬ 𝑥 ⊆ {∅} ↔ ∃𝑦(𝑦 ∈ 𝑥 ∧ ¬ 𝑦 ∈ {∅}))
6		simpll 779	. . . . . . . . . . . 12 ⊢ (((𝐹 ∈ (Fil‘∪ 𝐹) ∧ 𝑥 ⊆ (𝐹 ∪ {∅})) ∧ (𝑦 ∈ 𝑥 ∧ ¬ 𝑦 ∈ {∅})) → 𝐹 ∈ (Fil‘∪ 𝐹))
7		ssel2 3908	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ⊆ (𝐹 ∪ {∅}) ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ (𝐹 ∪ {∅}))
8	7	adantll 727	. . . . . . . . . . . . . . . 16 ⊢ (((𝐹 ∈ (Fil‘∪ 𝐹) ∧ 𝑥 ⊆ (𝐹 ∪ {∅})) ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ (𝐹 ∪ {∅}))
9		elun 4155	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ (𝐹 ∪ {∅}) ↔ (𝑦 ∈ 𝐹 ∨ 𝑦 ∈ {∅}))
10	8, 9	sylib 221	. . . . . . . . . . . . . . 15 ⊢ (((𝐹 ∈ (Fil‘∪ 𝐹) ∧ 𝑥 ⊆ (𝐹 ∪ {∅})) ∧ 𝑦 ∈ 𝑥) → (𝑦 ∈ 𝐹 ∨ 𝑦 ∈ {∅}))
11	10	orcomd 885	. . . . . . . . . . . . . 14 ⊢ (((𝐹 ∈ (Fil‘∪ 𝐹) ∧ 𝑥 ⊆ (𝐹 ∪ {∅})) ∧ 𝑦 ∈ 𝑥) → (𝑦 ∈ {∅} ∨ 𝑦 ∈ 𝐹))
12	11	ord 878	. . . . . . . . . . . . 13 ⊢ (((𝐹 ∈ (Fil‘∪ 𝐹) ∧ 𝑥 ⊆ (𝐹 ∪ {∅})) ∧ 𝑦 ∈ 𝑥) → (¬ 𝑦 ∈ {∅} → 𝑦 ∈ 𝐹))
13	12	impr 460	. . . . . . . . . . . 12 ⊢ (((𝐹 ∈ (Fil‘∪ 𝐹) ∧ 𝑥 ⊆ (𝐹 ∪ {∅})) ∧ (𝑦 ∈ 𝑥 ∧ ¬ 𝑦 ∈ {∅})) → 𝑦 ∈ 𝐹)
14		uniss 4878	. . . . . . . . . . . . . 14 ⊢ (𝑥 ⊆ (𝐹 ∪ {∅}) → ∪ 𝑥 ⊆ ∪ (𝐹 ∪ {∅}))
15	14	ad2antlr 740	. . . . . . . . . . . . 13 ⊢ (((𝐹 ∈ (Fil‘∪ 𝐹) ∧ 𝑥 ⊆ (𝐹 ∪ {∅})) ∧ (𝑦 ∈ 𝑥 ∧ ¬ 𝑦 ∈ {∅})) → ∪ 𝑥 ⊆ ∪ (𝐹 ∪ {∅}))
16		uniun 4893	. . . . . . . . . . . . . 14 ⊢ ∪ (𝐹 ∪ {∅}) = (∪ 𝐹 ∪ ∪ {∅})
17		0ex 5264	. . . . . . . . . . . . . . . 16 ⊢ ∅ ∈ V
18	17	unisn 4889	. . . . . . . . . . . . . . 15 ⊢ ∪ {∅} = ∅
19	18	uneq2i 4167	. . . . . . . . . . . . . 14 ⊢ (∪ 𝐹 ∪ ∪ {∅}) = (∪ 𝐹 ∪ ∅)
20		un0 4347	. . . . . . . . . . . . . 14 ⊢ (∪ 𝐹 ∪ ∅) = ∪ 𝐹
21	16, 19, 20	3eqtrri 2791	. . . . . . . . . . . . 13 ⊢ ∪ 𝐹 = ∪ (𝐹 ∪ {∅})
22	15, 21	sseqtrrdi 3954	. . . . . . . . . . . 12 ⊢ (((𝐹 ∈ (Fil‘∪ 𝐹) ∧ 𝑥 ⊆ (𝐹 ∪ {∅})) ∧ (𝑦 ∈ 𝑥 ∧ ¬ 𝑦 ∈ {∅})) → ∪ 𝑥 ⊆ ∪ 𝐹)
23		elssuni 4902	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ 𝑥 → 𝑦 ⊆ ∪ 𝑥)
24	23	ad2antrl 741	. . . . . . . . . . . 12 ⊢ (((𝐹 ∈ (Fil‘∪ 𝐹) ∧ 𝑥 ⊆ (𝐹 ∪ {∅})) ∧ (𝑦 ∈ 𝑥 ∧ ¬ 𝑦 ∈ {∅})) → 𝑦 ⊆ ∪ 𝑥)
25		filss 24187	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ (Fil‘∪ 𝐹) ∧ (𝑦 ∈ 𝐹 ∧ ∪ 𝑥 ⊆ ∪ 𝐹 ∧ 𝑦 ⊆ ∪ 𝑥)) → ∪ 𝑥 ∈ 𝐹)
26	6, 13, 22, 24, 25	syl13anc 1399	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ (Fil‘∪ 𝐹) ∧ 𝑥 ⊆ (𝐹 ∪ {∅})) ∧ (𝑦 ∈ 𝑥 ∧ ¬ 𝑦 ∈ {∅})) → ∪ 𝑥 ∈ 𝐹)
27		elun1 4183	. . . . . . . . . . 11 ⊢ (∪ 𝑥 ∈ 𝐹 → ∪ 𝑥 ∈ (𝐹 ∪ {∅}))
28	26, 27	syl 18	. . . . . . . . . 10 ⊢ (((𝐹 ∈ (Fil‘∪ 𝐹) ∧ 𝑥 ⊆ (𝐹 ∪ {∅})) ∧ (𝑦 ∈ 𝑥 ∧ ¬ 𝑦 ∈ {∅})) → ∪ 𝑥 ∈ (𝐹 ∪ {∅}))
29	28	ex 418	. . . . . . . . 9 ⊢ ((𝐹 ∈ (Fil‘∪ 𝐹) ∧ 𝑥 ⊆ (𝐹 ∪ {∅})) → ((𝑦 ∈ 𝑥 ∧ ¬ 𝑦 ∈ {∅}) → ∪ 𝑥 ∈ (𝐹 ∪ {∅})))
30	29	exlimdv 1967	. . . . . . . 8 ⊢ ((𝐹 ∈ (Fil‘∪ 𝐹) ∧ 𝑥 ⊆ (𝐹 ∪ {∅})) → (∃𝑦(𝑦 ∈ 𝑥 ∧ ¬ 𝑦 ∈ {∅}) → ∪ 𝑥 ∈ (𝐹 ∪ {∅})))
31	5, 30	biimtrid 245	. . . . . . 7 ⊢ ((𝐹 ∈ (Fil‘∪ 𝐹) ∧ 𝑥 ⊆ (𝐹 ∪ {∅})) → (¬ 𝑥 ⊆ {∅} → ∪ 𝑥 ∈ (𝐹 ∪ {∅})))
32		uni0b 4897	. . . . . . . 8 ⊢ (∪ 𝑥 = ∅ ↔ 𝑥 ⊆ {∅})
33		ssun2 4180	. . . . . . . . . 10 ⊢ {∅} ⊆ (𝐹 ∪ {∅})
34	17	snid 4623	. . . . . . . . . 10 ⊢ ∅ ∈ {∅}
35	33, 34	sselii 3910	. . . . . . . . 9 ⊢ ∅ ∈ (𝐹 ∪ {∅})
36		eleq1 2852	. . . . . . . . 9 ⊢ (∪ 𝑥 = ∅ → (∪ 𝑥 ∈ (𝐹 ∪ {∅}) ↔ ∅ ∈ (𝐹 ∪ {∅})))
37	35, 36	mpbiri 261	. . . . . . . 8 ⊢ (∪ 𝑥 = ∅ → ∪ 𝑥 ∈ (𝐹 ∪ {∅}))
38	32, 37	sylbir 238	. . . . . . 7 ⊢ (𝑥 ⊆ {∅} → ∪ 𝑥 ∈ (𝐹 ∪ {∅}))
39	31, 38	pm2.61d2 183	. . . . . 6 ⊢ ((𝐹 ∈ (Fil‘∪ 𝐹) ∧ 𝑥 ⊆ (𝐹 ∪ {∅})) → ∪ 𝑥 ∈ (𝐹 ∪ {∅}))
40	39	ex 418	. . . . 5 ⊢ (𝐹 ∈ (Fil‘∪ 𝐹) → (𝑥 ⊆ (𝐹 ∪ {∅}) → ∪ 𝑥 ∈ (𝐹 ∪ {∅})))
41	40	alrimiv 1961	. . . 4 ⊢ (𝐹 ∈ (Fil‘∪ 𝐹) → ∀𝑥(𝑥 ⊆ (𝐹 ∪ {∅}) → ∪ 𝑥 ∈ (𝐹 ∪ {∅})))
42		filin 24188	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (Fil‘∪ 𝐹) ∧ 𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝐹) → (𝑥 ∩ 𝑦) ∈ 𝐹)
43		elun1 4183	. . . . . . . . . 10 ⊢ ((𝑥 ∩ 𝑦) ∈ 𝐹 → (𝑥 ∩ 𝑦) ∈ (𝐹 ∪ {∅}))
44	42, 43	syl 18	. . . . . . . . 9 ⊢ ((𝐹 ∈ (Fil‘∪ 𝐹) ∧ 𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝐹) → (𝑥 ∩ 𝑦) ∈ (𝐹 ∪ {∅}))
45	44	3expa 1136	. . . . . . . 8 ⊢ (((𝐹 ∈ (Fil‘∪ 𝐹) ∧ 𝑥 ∈ 𝐹) ∧ 𝑦 ∈ 𝐹) → (𝑥 ∩ 𝑦) ∈ (𝐹 ∪ {∅}))
46	45	ralrimiva 3158	. . . . . . 7 ⊢ ((𝐹 ∈ (Fil‘∪ 𝐹) ∧ 𝑥 ∈ 𝐹) → ∀𝑦 ∈ 𝐹 (𝑥 ∩ 𝑦) ∈ (𝐹 ∪ {∅}))
47		elsni 4601	. . . . . . . . 9 ⊢ (𝑦 ∈ {∅} → 𝑦 = ∅)
48		ineq2 4114	. . . . . . . . . . 11 ⊢ (𝑦 = ∅ → (𝑥 ∩ 𝑦) = (𝑥 ∩ ∅))
49		in0 4348	. . . . . . . . . . 11 ⊢ (𝑥 ∩ ∅) = ∅
50	48, 49	eqtrdi 2814	. . . . . . . . . 10 ⊢ (𝑦 = ∅ → (𝑥 ∩ 𝑦) = ∅)
51	50, 35	eqeltrdi 2872	. . . . . . . . 9 ⊢ (𝑦 = ∅ → (𝑥 ∩ 𝑦) ∈ (𝐹 ∪ {∅}))
52	47, 51	syl 18	. . . . . . . 8 ⊢ (𝑦 ∈ {∅} → (𝑥 ∩ 𝑦) ∈ (𝐹 ∪ {∅}))
53	52	rgen 3082	. . . . . . 7 ⊢ ∀𝑦 ∈ {∅} (𝑥 ∩ 𝑦) ∈ (𝐹 ∪ {∅})
54		ralun 4199	. . . . . . 7 ⊢ ((∀𝑦 ∈ 𝐹 (𝑥 ∩ 𝑦) ∈ (𝐹 ∪ {∅}) ∧ ∀𝑦 ∈ {∅} (𝑥 ∩ 𝑦) ∈ (𝐹 ∪ {∅})) → ∀𝑦 ∈ (𝐹 ∪ {∅})(𝑥 ∩ 𝑦) ∈ (𝐹 ∪ {∅}))
55	46, 53, 54	sylancl 598	. . . . . 6 ⊢ ((𝐹 ∈ (Fil‘∪ 𝐹) ∧ 𝑥 ∈ 𝐹) → ∀𝑦 ∈ (𝐹 ∪ {∅})(𝑥 ∩ 𝑦) ∈ (𝐹 ∪ {∅}))
56	55	ralrimiva 3158	. . . . 5 ⊢ (𝐹 ∈ (Fil‘∪ 𝐹) → ∀𝑥 ∈ 𝐹 ∀𝑦 ∈ (𝐹 ∪ {∅})(𝑥 ∩ 𝑦) ∈ (𝐹 ∪ {∅}))
57		elsni 4601	. . . . . . 7 ⊢ (𝑥 ∈ {∅} → 𝑥 = ∅)
58		ineq1 4113	. . . . . . . . . 10 ⊢ (𝑥 = ∅ → (𝑥 ∩ 𝑦) = (∅ ∩ 𝑦))
59		0in 4350	. . . . . . . . . 10 ⊢ (∅ ∩ 𝑦) = ∅
60	58, 59	eqtrdi 2814	. . . . . . . . 9 ⊢ (𝑥 = ∅ → (𝑥 ∩ 𝑦) = ∅)
61	60, 35	eqeltrdi 2872	. . . . . . . 8 ⊢ (𝑥 = ∅ → (𝑥 ∩ 𝑦) ∈ (𝐹 ∪ {∅}))
62	61	ralrimivw 3162	. . . . . . 7 ⊢ (𝑥 = ∅ → ∀𝑦 ∈ (𝐹 ∪ {∅})(𝑥 ∩ 𝑦) ∈ (𝐹 ∪ {∅}))
63	57, 62	syl 18	. . . . . 6 ⊢ (𝑥 ∈ {∅} → ∀𝑦 ∈ (𝐹 ∪ {∅})(𝑥 ∩ 𝑦) ∈ (𝐹 ∪ {∅}))
64	63	rgen 3082	. . . . 5 ⊢ ∀𝑥 ∈ {∅}∀𝑦 ∈ (𝐹 ∪ {∅})(𝑥 ∩ 𝑦) ∈ (𝐹 ∪ {∅})
65		ralun 4199	. . . . 5 ⊢ ((∀𝑥 ∈ 𝐹 ∀𝑦 ∈ (𝐹 ∪ {∅})(𝑥 ∩ 𝑦) ∈ (𝐹 ∪ {∅}) ∧ ∀𝑥 ∈ {∅}∀𝑦 ∈ (𝐹 ∪ {∅})(𝑥 ∩ 𝑦) ∈ (𝐹 ∪ {∅})) → ∀𝑥 ∈ (𝐹 ∪ {∅})∀𝑦 ∈ (𝐹 ∪ {∅})(𝑥 ∩ 𝑦) ∈ (𝐹 ∪ {∅}))
66	56, 64, 65	sylancl 598	. . . 4 ⊢ (𝐹 ∈ (Fil‘∪ 𝐹) → ∀𝑥 ∈ (𝐹 ∪ {∅})∀𝑦 ∈ (𝐹 ∪ {∅})(𝑥 ∩ 𝑦) ∈ (𝐹 ∪ {∅}))
67		p0ex 5350	. . . . . 6 ⊢ {∅} ∈ V
68		unexg 7770	. . . . . 6 ⊢ ((𝐹 ∈ (Fil‘∪ 𝐹) ∧ {∅} ∈ V) → (𝐹 ∪ {∅}) ∈ V)
69	67, 68	mpan2 704	. . . . 5 ⊢ (𝐹 ∈ (Fil‘∪ 𝐹) → (𝐹 ∪ {∅}) ∈ V)
70		istopg 23228	. . . . 5 ⊢ ((𝐹 ∪ {∅}) ∈ V → ((𝐹 ∪ {∅}) ∈ Top ↔ (∀𝑥(𝑥 ⊆ (𝐹 ∪ {∅}) → ∪ 𝑥 ∈ (𝐹 ∪ {∅})) ∧ ∀𝑥 ∈ (𝐹 ∪ {∅})∀𝑦 ∈ (𝐹 ∪ {∅})(𝑥 ∩ 𝑦) ∈ (𝐹 ∪ {∅}))))
71	69, 70	syl 18	. . . 4 ⊢ (𝐹 ∈ (Fil‘∪ 𝐹) → ((𝐹 ∪ {∅}) ∈ Top ↔ (∀𝑥(𝑥 ⊆ (𝐹 ∪ {∅}) → ∪ 𝑥 ∈ (𝐹 ∪ {∅})) ∧ ∀𝑥 ∈ (𝐹 ∪ {∅})∀𝑦 ∈ (𝐹 ∪ {∅})(𝑥 ∩ 𝑦) ∈ (𝐹 ∪ {∅}))))
72	41, 66, 71	mpbir2and 726	. . 3 ⊢ (𝐹 ∈ (Fil‘∪ 𝐹) → (𝐹 ∪ {∅}) ∈ Top)
73	21	cldopn 23364	. . . . . . . 8 ⊢ (𝑥 ∈ (Clsd‘(𝐹 ∪ {∅})) → (∪ 𝐹 ∖ 𝑥) ∈ (𝐹 ∪ {∅}))
74		elun 4155	. . . . . . . 8 ⊢ ((∪ 𝐹 ∖ 𝑥) ∈ (𝐹 ∪ {∅}) ↔ ((∪ 𝐹 ∖ 𝑥) ∈ 𝐹 ∨ (∪ 𝐹 ∖ 𝑥) ∈ {∅}))
75	73, 74	sylib 221	. . . . . . 7 ⊢ (𝑥 ∈ (Clsd‘(𝐹 ∪ {∅})) → ((∪ 𝐹 ∖ 𝑥) ∈ 𝐹 ∨ (∪ 𝐹 ∖ 𝑥) ∈ {∅}))
76		elun 4155	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝐹 ∪ {∅}) ↔ (𝑥 ∈ 𝐹 ∨ 𝑥 ∈ {∅}))
77		filfbas 24182	. . . . . . . . . . . . . 14 ⊢ (𝐹 ∈ (Fil‘∪ 𝐹) → 𝐹 ∈ (fBas‘∪ 𝐹))
78		fbncp 24173	. . . . . . . . . . . . . 14 ⊢ ((𝐹 ∈ (fBas‘∪ 𝐹) ∧ 𝑥 ∈ 𝐹) → ¬ (∪ 𝐹 ∖ 𝑥) ∈ 𝐹)
79	77, 78	sylan 592	. . . . . . . . . . . . 13 ⊢ ((𝐹 ∈ (Fil‘∪ 𝐹) ∧ 𝑥 ∈ 𝐹) → ¬ (∪ 𝐹 ∖ 𝑥) ∈ 𝐹)
80	79	pm2.21d 122	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ (Fil‘∪ 𝐹) ∧ 𝑥 ∈ 𝐹) → ((∪ 𝐹 ∖ 𝑥) ∈ 𝐹 → 𝑥 = ∅))
81	80	ex 418	. . . . . . . . . . 11 ⊢ (𝐹 ∈ (Fil‘∪ 𝐹) → (𝑥 ∈ 𝐹 → ((∪ 𝐹 ∖ 𝑥) ∈ 𝐹 → 𝑥 = ∅)))
82	57	a1i13 28	. . . . . . . . . . 11 ⊢ (𝐹 ∈ (Fil‘∪ 𝐹) → (𝑥 ∈ {∅} → ((∪ 𝐹 ∖ 𝑥) ∈ 𝐹 → 𝑥 = ∅)))
83	81, 82	jaod 873	. . . . . . . . . 10 ⊢ (𝐹 ∈ (Fil‘∪ 𝐹) → ((𝑥 ∈ 𝐹 ∨ 𝑥 ∈ {∅}) → ((∪ 𝐹 ∖ 𝑥) ∈ 𝐹 → 𝑥 = ∅)))
84	76, 83	biimtrid 245	. . . . . . . . 9 ⊢ (𝐹 ∈ (Fil‘∪ 𝐹) → (𝑥 ∈ (𝐹 ∪ {∅}) → ((∪ 𝐹 ∖ 𝑥) ∈ 𝐹 → 𝑥 = ∅)))
85	84	imp 412	. . . . . . . 8 ⊢ ((𝐹 ∈ (Fil‘∪ 𝐹) ∧ 𝑥 ∈ (𝐹 ∪ {∅})) → ((∪ 𝐹 ∖ 𝑥) ∈ 𝐹 → 𝑥 = ∅))
86		elsni 4601	. . . . . . . . 9 ⊢ ((∪ 𝐹 ∖ 𝑥) ∈ {∅} → (∪ 𝐹 ∖ 𝑥) = ∅)
87		elssuni 4902	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (𝐹 ∪ {∅}) → 𝑥 ⊆ ∪ (𝐹 ∪ {∅}))
88	87, 21	sseqtrrdi 3954	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (𝐹 ∪ {∅}) → 𝑥 ⊆ ∪ 𝐹)
89	88	adantl 487	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (Fil‘∪ 𝐹) ∧ 𝑥 ∈ (𝐹 ∪ {∅})) → 𝑥 ⊆ ∪ 𝐹)
90		ssdif0 4317	. . . . . . . . . . 11 ⊢ (∪ 𝐹 ⊆ 𝑥 ↔ (∪ 𝐹 ∖ 𝑥) = ∅)
91	90	biimpri 231	. . . . . . . . . 10 ⊢ ((∪ 𝐹 ∖ 𝑥) = ∅ → ∪ 𝐹 ⊆ 𝑥)
92		eqss 3928	. . . . . . . . . . 11 ⊢ (𝑥 = ∪ 𝐹 ↔ (𝑥 ⊆ ∪ 𝐹 ∧ ∪ 𝐹 ⊆ 𝑥))
93	92	simplbi2 506	. . . . . . . . . 10 ⊢ (𝑥 ⊆ ∪ 𝐹 → (∪ 𝐹 ⊆ 𝑥 → 𝑥 = ∪ 𝐹))
94	89, 91, 93	syl2im 41	. . . . . . . . 9 ⊢ ((𝐹 ∈ (Fil‘∪ 𝐹) ∧ 𝑥 ∈ (𝐹 ∪ {∅})) → ((∪ 𝐹 ∖ 𝑥) = ∅ → 𝑥 = ∪ 𝐹))
95	86, 94	syl5 35	. . . . . . . 8 ⊢ ((𝐹 ∈ (Fil‘∪ 𝐹) ∧ 𝑥 ∈ (𝐹 ∪ {∅})) → ((∪ 𝐹 ∖ 𝑥) ∈ {∅} → 𝑥 = ∪ 𝐹))
96	85, 95	orim12d 979	. . . . . . 7 ⊢ ((𝐹 ∈ (Fil‘∪ 𝐹) ∧ 𝑥 ∈ (𝐹 ∪ {∅})) → (((∪ 𝐹 ∖ 𝑥) ∈ 𝐹 ∨ (∪ 𝐹 ∖ 𝑥) ∈ {∅}) → (𝑥 = ∅ ∨ 𝑥 = ∪ 𝐹)))
97	75, 96	syl5 35	. . . . . 6 ⊢ ((𝐹 ∈ (Fil‘∪ 𝐹) ∧ 𝑥 ∈ (𝐹 ∪ {∅})) → (𝑥 ∈ (Clsd‘(𝐹 ∪ {∅})) → (𝑥 = ∅ ∨ 𝑥 = ∪ 𝐹)))
98	97	expimpd 459	. . . . 5 ⊢ (𝐹 ∈ (Fil‘∪ 𝐹) → ((𝑥 ∈ (𝐹 ∪ {∅}) ∧ 𝑥 ∈ (Clsd‘(𝐹 ∪ {∅}))) → (𝑥 = ∅ ∨ 𝑥 = ∪ 𝐹)))
99		elin 4096	. . . . 5 ⊢ (𝑥 ∈ ((𝐹 ∪ {∅}) ∩ (Clsd‘(𝐹 ∪ {∅}))) ↔ (𝑥 ∈ (𝐹 ∪ {∅}) ∧ 𝑥 ∈ (Clsd‘(𝐹 ∪ {∅}))))
100		vex 3458	. . . . . 6 ⊢ 𝑥 ∈ V
101	100	elpr 4609	. . . . 5 ⊢ (𝑥 ∈ {∅, ∪ 𝐹} ↔ (𝑥 = ∅ ∨ 𝑥 = ∪ 𝐹))
102	98, 99, 101	3imtr4g 299	. . . 4 ⊢ (𝐹 ∈ (Fil‘∪ 𝐹) → (𝑥 ∈ ((𝐹 ∪ {∅}) ∩ (Clsd‘(𝐹 ∪ {∅}))) → 𝑥 ∈ {∅, ∪ 𝐹}))
103	102	ssrdv 3919	. . 3 ⊢ (𝐹 ∈ (Fil‘∪ 𝐹) → ((𝐹 ∪ {∅}) ∩ (Clsd‘(𝐹 ∪ {∅}))) ⊆ {∅, ∪ 𝐹})
104	21	isconn2 23747	. . 3 ⊢ ((𝐹 ∪ {∅}) ∈ Conn ↔ ((𝐹 ∪ {∅}) ∈ Top ∧ ((𝐹 ∪ {∅}) ∩ (Clsd‘(𝐹 ∪ {∅}))) ⊆ {∅, ∪ 𝐹}))
105	72, 103, 104	sylanbrc 595	. 2 ⊢ (𝐹 ∈ (Fil‘∪ 𝐹) → (𝐹 ∪ {∅}) ∈ Conn)
106	4, 105	syl 18	1 ⊢ (𝐹 ∈ (Fil‘𝑋) → (𝐹 ∪ {∅}) ∈ Conn)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   ∧ w3a 1103  ∀wal 1567   = wceq 1569  ∃wex 1811   ∈ wcel 2146  ∀wral 3080  Vcvv 3454   ⊆ wss 3900   ∖ cdif 4048   ∩ cin 4092   ∪ cun 4152  ∅c0 4281  {csn 4588  {cpr 4590  ∪ cuni 4870  ‘cfv 6547  fBascfbas 21681  Topctop 23226  Clsdccld 23349  Conncconn 23744  Filcfil 24179

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2196  ax-12 2215  ax-ext 2735  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7761

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2101  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5171  df-mpt 5190  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6501  df-fun 6549  df-fn 6550  df-fv 6555  df-fbas 21690  df-top 23227  df-cld 23352  df-conn 23745  df-fil 24180

This theorem is used by:  ufildr  24265